Théorie Financière Structure financière et coût du capital Professeur André Farber

Risk and return and capital budgeting Objectives for this session: Beta of a portfolio Beta and leverage Weighted average cost of capital Modigliani Miller 1958 WACC with taxes Tfin 2005 11 Intro to capital structure

Beta of a portfolio Consider the following portfolio Stock Value Beta A nA  PA A B nB  PB B The value of the portfolio is: V = nA  PA + nA  PB The fractions invested in each stock are: Xi = ( ni Pi) / V for i = A,B The beta of the portfolio is the weighted average of the betas of the individual stocks P = XA A + XB B Example 1 Stock $  X ATT 3,000 0,76 0.60 GENENTECH 2,000 1.40 0.40 V 5,000 P = 0.60  0.76 + 0.40  1.40 = 1.02 Tfin 2005 11 Intro to capital structure

Example Stock $  X ATT 3,000 0.76 0.60 Genetech 2,000 1.40 0.40 V 5,000 P = 0.60 * 0.76 + 0.40 * 1.40 = 1.02 Tfin 2005 11 Intro to capital structure

Application 1: cost of capital for a division Firm = collection of assets Example: A company has two divisions Value($ mio)  Electrical 100 0.50 Chemical 500 0.90 V 600 firm = (100/600) * (0.50) + (500/600) * 0.90 = 0.83 Assume: rf = 5% rM - rf = 6% Expected return on stocks: r = 5% + 6%  0.83 = 9.98 % An adequate hurdle rate for capital budgeting decisions ? No The firm should use required rate of returns based on project risks: Electricity : 5 + 6  0.50 = 8% Chemical : 5 + 6  0.90 = 10.4% Tfin 2005 11 Intro to capital structure

Application 2: leverage and beta Consider an investor who borrows at the risk free rate to invest in the market portfolio Assets $  X Market portfolio 2,000 1 2 Risk-free rate -1,000 0 -1 V 1,000 P = 2  1 + (-1)  0 = 2 Tfin 2005 11 Intro to capital structure

Expected Return Expected Return P 20% P M 14% 14% M 8% 8% 2 Sigma 1 Beta Tfin 2005 11 Intro to capital structure

Cost of capital with debt Up to now, the analysis has proceeded based on the assumption that investment decisions are independent of financing decisions. Does the value of a company change the cost of capital change if leverage changes ? Tfin 2005 11 Intro to capital structure

An example CAPM holds – Risk-free rate = 5%, Market risk premium = 6% Consider an all-equity firm: Market value V 100 Beta 1 Cost of capital 11% (=5% + 6% * 1) Now consider borrowing 20 to buy back shares. Why such a move? Debt is cheaper than equity Replacing equity with debt should reduce the average cost of financing What will be the final impact On the value of the company? (Equity + Debt)? On the weighted average cost of capital (WACC)? Tfin 2005 11 Intro to capital structure

Weighted Average Cost of Capital An average of: The cost of equity requity The cost of debt rdebt Weighted by their relative market values (E/V and D/V) Note: V = E + D Tfin 2005 11 Intro to capital structure

Modigliani Miller (1958) Assume perfect capital markets: not taxes, no transaction costs Proposition I: The market value of any firm is independent of its capital structure: V = E+D = VU Proposition II: The weighted average cost of capital is independent of its capital structure rwacc = rA rA is the cost of capital of an all equity firm Tfin 2005 11 Intro to capital structure

Using MM 58 Value of company: V = 100 Initial Final Equity 100 80 Debt 0 20 Total 100 100 MM I WACC = rA 11% 11% MM II Cost of debt - 5% (assuming risk-free debt) D/V 0 0.20 Cost of equity 11% 12.50% (to obtain rwacc = 11%) E/V 100% 80% Tfin 2005 11 Intro to capital structure

Cost of equity calculation V (=VU ) = E + D Value of equity rE Value of all-equity firm rA rD Value of debt Tfin 2005 11 Intro to capital structure

Why is rwacc unchanged? Consider someone owning a portfolio of all firm’s securities (debt and equity) with Xequity = E/V (80% in example ) and Xdebt = D/V (20%) Expected return on portfolio = requity * Xequity + rdebt * Xdebt This is equal to the WACC (see definition): rportoflio = rwacc But she/he would, in fact, own a fraction of the company. The expected return would be equal to the expected return of the unlevered (all equity) firm rportoflio = rA The weighted average cost of capital is thus equal to the cost of capital of an all equity firm rwacc = rA Tfin 2005 11 Intro to capital structure

What are MM I and MM II related? Assumption: perpetuities (to simplify the presentation) For a levered companies, earnings before interest and taxes will be split between interest payments and dividends payments EBIT = Int + Div Market value of equity: present value of future dividends discounted at the cost of equity E = Div / requity Market value of debt: present value of future interest discounted at the cost of debt D = Int / rdebt Tfin 2005 11 Intro to capital structure

Relationship between the value of company and WACC From the definition of the WACC: rwacc * V = requity * E + rdebt * D As requity * E = Div and rdebt * D = Int rwacc * V = EBIT V = EBIT / rwacc Market value of levered firm EBIT is independent of leverage If value of company varies with leverage, so does WACC in opposite direction Tfin 2005 11 Intro to capital structure

MM II: another presentation The equality rwacc = rA can be written as: Expected return on equity is an increasing function of leverage: requity 12.5% Additional cost due to leverage 11% rA rwacc 5% rdebt D/E 0.25 Tfin 2005 11 Intro to capital structure

Why does requity increases with leverage? Because leverage increases the risk of equity. To see this, back to the portfolio with both debt and equity. Beta of portfolio: portfolio = equity * Xequity + debt * Xdebt But also: portfolio = Asset So: or Tfin 2005 11 Intro to capital structure

Back to example Assume debt is riskless: Tfin 2005 11 Intro to capital structure

Summary: the Beta-CAPM diagram Executive Master in Finance – Modigliani Miller 1958 Summary: the Beta-CAPM diagram   Beta L βEquity U βAsset r rEquity rAsset rDebt=rf D/E The IS-LM diagram used by economist is the source of inspiration for this figure. For many years, I have tried to create a figure which would look as impressive as the IS-LM diagram. Here is the result. The figure in the first quadrant shows the relationship between the debt-equity ratio and the beta equity. The figure is based on the assumption that the debt is riskless. In that case, the relationship is linear. The security market line in illustrated in the second quadrant.  The relationship between the debt-equity ratio and the cost of equity, the cost of debt and the WACC is in the third quadrant  This is simply a -45% line. This diagram can be used to show that investors can choose the level of leverage that they wish by rebalancing their portfolios. If the company has no debt (point U), an investor can create leverage by borrowing at the risk-free interest to reach L, the risk and expected returns of a levered company. If, on the hand, the company is levered (point L), an investor can allocate his money between the stock of the levered company and the risk free asset to undo the leverage and reach point U.   rEquity rDebt D/E WACC Executive Master of Finance 2007 01 Modigliani Miller

Risky debt: Merton model Limited liability: rules out negative equity Market value of equity Equity similar to a call option on the company Company goes bankrupt if V<F Market value of company V Face value of debt F Tfin 2005 11 Intro to capital structure

Risk debt decomposition Risky debt = Risk-free debt - Put Market value of debt Bankruptcy Face value of debt Market value of company Face value of debt Tfin 2005 11 Intro to capital structure

Figure 21.9 Beta of Debt and Equity The blue curve is the beta of equity and the red curve is the beta of debt as a function of the firm’s debt-to-equity ratio. The black line is the approximation derived in Chapter 14—the beta of equity when the beta of debt is assumed to be zero. The firm is assumed to hold five-year zero-coupon debt and reinvest all its earnings. (The firm’s beta of assets is 1, the risk-free interest rate is 3% per year, and the volatility of assets is 20% per year.) Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

Corporate Tax Shield Interest are tax deductible => tax shield Tax shield = Interest payment × Corporate Tax Rate = (rD × D) × TC rD : cost of new debt D : market value of debt Value of levered firm = Value if all-equity-financed + PV(Tax Shield) PV(Tax Shield) - Assume permanent borrowing VL=VU + TCD Tfin 2005 11 Intro to capital structure

Cost of equity calculation VU + T = E + D Value of equity rA rE Value of all-equity firm rD Value of debt rD Value of tax shield Tfin 2005 11 Intro to capital structure

Cost of equity calculation (2) General formula If rTS = rD & VTS = TCD Similar formulas for beta equity (replace r by β) Tfin 2005 11 Intro to capital structure

WACC If g =0 (constant perpetuity case): VTS = TCD wacc = rA(1-TC D/VL) This is the MM formula. Tfin 2005 11 Intro to capital structure

Example A B Balance Sheet Total Assets 1,000 1,000 Book Equity 1,000 500 Debt (8%) 0 500 Income Statement EBIT 240 240 Interest 0 40 Taxable Income 240 200 Taxes (40%) 96 80 Net Income 144 120 Dividend 144 120 Total 144 160 Assume rA= 10% (1) Value of all-equity-firm: VU = 144 / 0.10 = 1,440 (2) PV(Tax Shield): Tax Shield = 40 x 0.40 = 16 PV(TaxShield) = 16/0.08 = 200 (3) Value of levered company: VL = 1,440 + 200 = 1,640 (5) Market value of equity: EL = VL - D = 1,640 - 500 = 1,140 Tfin 2005 11 Intro to capital structure

What about cost of equity? Proof: But VU = EBIT(1-TC)/rA and E = VU + TCD – D Replace and solve 1) Cost of equity increases with leverage: 2) Beta of equity increases In example: rE = 10% +(10%-8%)(1-0.4)(500/1,140) = 10.53% or rE = DIV/E = 120/1,140 = 10.53% Tfin 2005 11 Intro to capital structure

What about the weighted average cost of capital? Weighted average cost of capital decreases with leverage Weighted average cost of capital: discount rate used to calculate the market value of firm by discounting net operating profit less adjusted taxes (NOPLAT) NOPLAT = Net Income + Interest + Tax Shield = (EBIT-rDD)(1-TC) + rDD +TCrDD = Net Income for all-equity-firm = EBIT(1-TC) VL = NOPLAT / WACC As: In example: NOPLAT = 144 VL = 1,640 WACC = 10.53% x 0.69 + 8% x 0.60 x 0.31 = 8.78% Tfin 2005 11 Intro to capital structure